Systems and methods for applying scaling laws of tree structures

ABSTRACT

In at least one embodiment, a method for determining the resistance of a flow within at least a portion of a vessel is provided, with the method comprising the steps of obtaining a biological tree image showing a structure of at least part of a biological tree, identifying a length of a vessel portion and a diameter of a stem vessel proximal to the vessel crown from the biological tree image, and calculating resistance based upon at least the length of a vessel portion and a diameter of a stem vessel proximal to the vessel crown. In another embodiment, a method for determining vessel volume is provided, with the method comprising the steps of obtaining a biological tree image showing a structure of at least part of a biological tree, identifying a diameter of a stem segment and a length of a stem segment from the biological tree image, and calculating cumulative crown volume based upon at least the diameter of the stem segment and the length of the stem segment.

PRIORITY

The present application is related to and claims the benefit ofInternational Patent Application Serial No. PCT/US2008/000762, entitled“APPLICATIONS OF SCALING LAWS OF TREE STRUCTURES,” filed Jan. 22, 2008,and U.S. Provisional Patent Application Ser. No. 60/881,833, entitled“APPLICATIONS OF SCALING LAWS OF TREE STRUCTURES,” filed Jan. 23, 2007,the contents of which are hereby incorporated by reference.

BACKGROUND

The disclosure of the present application relates generally to diagnosisof vascular disease, in particular relating to using morphologicalfeatures of the coronary artery tree to diagnose coronary arterydisease.

Diffuse coronary artery disease (DCAD), a common form ofatherosclerosis, is difficult to diagnose because the arterial lumencross-sectional area is diffusely reduced along the length of thevessels. Typically, for patients with even mild segmental stenosis, thelumen cross-sectional area is diffusely reduced by 30 to 50%. Thefailure of improved coronary flow reserve after angioplasty may mainlybe due to the coexistence of diffuse narrowing and focal stenosis.Whereas angiography has been regarded as the “gold standard” in theassessment of focal stenosis of coronary arteries, its viability todiagnose DCAD remains questionable. The rationale of conventionalangiography in the assessment of coronary artery disease is to calculatethe percent lumen diameter reduction by comparison of the target segmentwith the adjacent ‘normal’ reference segment. In the presence of DCAD,however, an entire vessel may be diffusely narrowed so that no truereference (normal) segment exists. Therefore, in the presence of DCAD,standard angiography significantly underestimates the severity of thedisease.

To overcome the difficulty of using angiography in the diagnosis ofDCAD, intravascular ultrasound (IVUS) has been the subject of extensivestudies. IVUS has the advantage of directly imaging the cross-sectionalarea along the length of the vessel using a small catheter. Thedisadvantage of IVUS, however, is that its extensive interrogation ofdiseased segments may pose a risk for plaque rupture.

In addition to the foregoing, biological transport structures (vasculartrees, for example), have significant similarities despite remarkablediversity and size across species. The vascular tree, whose function isto transport fluid within an organism, is a major distribution system,which has known fractal and scaling characteristics. A fundamentalfunctional parameter of a vessel segment or a tree is the hydraulicresistance to flow, which determines the transport efficiency. It isimportant to understand the hydraulic resistance of a vascular treebecause it is the major determinant of transport in biology.

In a hydrodynamic analysis of mammalian and plant vascular networks, amathematical model of ¾-power scaling for metabolic rates has beenreported. A number of scaling relations of structure-function featureswere further proposed for body size, temperature, species abundance,body growth, and so on. Although the ¾ scaling law was originallyderived through a hemodynamic analysis in the vascular tree system, atleast one basic structure-function scaling feature of vascular treesremains unclear: “How does the resistance of a vessel branch scale withthe equivalent resistance of the corresponding distal tree?”

What is needed is an improved approach to diagnosis and prognosis ofvascular disease and its symptoms that avoid intrusive and expensivemethods while improving accuracy and efficacy. Such an approach mayinclude, for example, a novel scaling law of a single vessel resistanceas relative to its corresponding distal tree.

Blood pressure and perfusion of an organ depend on a complex interplaybetween cardiac output, intravascular volume, and vasomotor tone,amongst others. The vascular system provides the basic architecture totransport the fluids while other physical, physiological, and chemicalfactors affect the intravascular volume to regulate the flow in thebody. Although the intravascular volume can adapt to normal physicaltraining, many diagnostic and treatment options depend on the estimationof the volume status of patients. For example, a recent study classifiedblood volume status as hypovolemic, normovolemic, and hypervolemic.

Heart failure results in an increase of intravascular volume(hypervolemia) in response to decreased cardiac output and renalhypoperfusion. Conversely, myocardial ischemia and infarct lead to adecrease of intravascular volume (hypovolemia) distal to an occludedcoronary artery, and patients with postural tachycardia syndrome alsoshow hypervolemia. Furthermore, patients of edematous disorders havebeen found to have abnormal blood volume. Currently, there is nononinvasive method to determine the blood volume in sub-organ, organs,organ system or organism. The disclosure of the present applicationprovides a novel scaling law that provides the basis for determinationof blood volume throughout the vasculature.

BRIEF SUMMARY

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the method comprises the steps of obtaining abiological tree image showing a structure of at least part of abiological tree, identifying a length of a vessel portion and a diameterof a stem vessel proximal to the vessel crown from the biological treeimage, and calculating resistance based upon at least the length of avessel portion and a diameter of a stem vessel proximal to the vesselcrown. In another embodiment, the step of calculating resistance isfurther based upon the use of a constant. In yet another embodiment, theresistance is a resistance of a stem segment, and wherein the length isa length of a stem. In an additional embodiment, the step of calculatingthe resistance of a stem segment is further based upon the use of aconstant for the stem. In yet an additional embodiment, the constant forthe stem is equivalent to one hundred and twenty eight multiplied by afluid viscosity divided by pi, or a mathematical equivalent thereof.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the step of calculating the resistance of a stemsegment is performed by dividing the length of a stem by the diameter ofa stem vessel proximal to the vessel crown to the fourth powermultiplied by a constant for the stem, or a mathematical equivalentthereof. In another embodiment, the constant for the stem is equivalentto one hundred and twenty eight multiplied by a fluid viscosity dividedby pi, or a mathematical equivalent thereof. In yet another embodiment,the resistance is a resistance of a crown, and *herein the length is acrown length. In an additional embodiment, the crown length isdetermined by a sum of lengths of all or substantially all of thevessels in a crown. In yet an additional embodiment, the step ofcalculating the resistance of a crown is further based upon the use of aconstant for the crown.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the constant for the crown is equivalent to amaximum resistance multiplied by a maximum diameter to the fourth powerdivided by a maximum length, or a mathematical equivalent thereof. Inanother embodiment, the step of calculating the resistance of a crown isperformed by dividing the crown length by the diameter of a stem vesselproximal to the vessel crown to the fourth power multiplied by aconstant, or a mathematical equivalent thereof. In yet anotherembodiment, the constant for the crown is equivalent to a maximumresistance multiplied by a maximum diameter to the fourth power dividedby a maximum length, or a mathematical equivalent thereof. In anadditional embodiment, in the constant for the crown is dependent upon abranching ratio, a diameter ratio, a total number of tree generations,and viscosity within a crown. In yet an additional embodiment, theresistance is a total resistance of the biological tree, and Wherein thelength is a cumulative biological tree vessel length.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the step of calculating the total resistance of thebiological tree is further based upon the use of a constant for thecrown. In another embodiment, the constant for the crown is equivalentto the total resistance of the biological tree multiplied by a mostproximal stem diameter to the fourth power divided by a cumulativebiological tree vessel length., or a mathematical equivalent thereof. Inyet another embodiment, the step of calculating the total resistance ofthe biological tree is performed by dividing the cumulative biologicaltree vessel length by the diameter of a stem vessel proximal to thevessel crown to the fourth power multiplied by a constant, or amathematical equivalent thereof. In an additional embodiment, theconstant for the crown is equivalent to the total resistance of thebiological tree multiplied by a most proximal stem diameter to thefourth power divided by a cumulative biological tree vessel length, or amathematical equivalent thereof. In yet an additional embodiment, theresistance is the resistance of blood within a vessel portion.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the vessel portion comprises a vessel selected fromthe group consisting of a blood vessel, a bile duct, a lung, and aneuron. In another embodiment, the resistance is the resistance of a gaswithin a vessel portion. In yet another embodiment, the gas comprisesbreathable air, and wherein the vessel portion comprises a portion of alung. In an additional embodiment, the method further comprises the stepof comparing the calculated resistance to a calculated model resistanceto determine the extent of vessel and/or organ disease. In yet anadditional embodiment, the step of comparing the calculated resistanceto the calculated model resistance is performed by graphically comparingcalculated resistance data to model resistance calculation data todetermine the extent of vessel and/or organ disease by identifying oneor more graphical differences between said data.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the step of comparing the calculated resistance tothe calculated model resistance is performed by comparing calculatedresistance data to model resistance calculation data in table form todetermine the extent of vessel and/or organ disease by identifying oneor more numerically calculated differences between said data. In anotherembodiment, the resistance is a resistance of a stem segment, andwherein the length is a length of a stem, and the method furthercomprises the steps of identifying a crown length from the biologicaltree image, and calculating crown resistance based upon at least thelength of a vessel portion and a diameter of a stem vessel proximal tothe vessel crown. In yet another embodiment, the method furthercomprises the step of determining a resistance scaling relation, whereinthe resistance scaling relation comprises a first component, a secondcomponent, and a third component, or a mathematical equivalent thereof.In an additional embodiment, the first component comprises theresistance of a stem segment divided by the crown resistance, whereinthe second component comprises a constant for the stem divided by aconstant for the crown, and wherein the third component comprises thelength of a stem divided by the crown length, and wherein the firstcomponent equals the second component multiplied by the third component,or a mathematical equivalent thereof. In yet an additional embodiment,the resistance is a resistance of a crown, the length is a length of astem, and the method further comprises the steps of identifying acumulative biological tree vessel length from the biological tree image,identifying a most proximal stem diameter from the biological treeimage, and calculating a total resistance of at least part of abiological tree based upon at least the cumulative biological treevessel length and the most proximal stem diameter.

In at least one embodiment of a method for determining the resistance ofa flow within at least a portion of a vessel of the disclosure of thepresent application, the step of calculating a total resistance furthercomprises the use of a known parameter. In another embodiment, themethod further comprises the step of determining a resistance scalingrelation, wherein the resistance scaling relation comprises a firstcomponent, a second component, and a third component, or a mathematicalequivalent thereof. In yet another embodiment, the first componentcomprises the resistance of a crown divided by the total resistance,wherein the second component comprise the diameter of a stem vesselproximal to the vessel portion divided by the most proximal stemdiameter, and wherein the third component comprises the length of acrown divided by the cumulative biological tree vessel length, andwherein the first component multiplied by the second component to thefourth power equals a known parameter multiplied by the third component,or a mathematical equivalent thereof. In an additional embodiment, thecalculated resistance provides information useful for a diagnosis of adisease.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the method comprises the steps of obtaining amodel biological tree, obtaining a patient's biological tree imageshowing a vasculature of at least part of a patient's biological tree,identifying a length of a patient vessel portion and a diameter of apatient stem vessel proximal to the vessel crown from the patient'sbiological tree image, calculating model resistance based upon at leasta length of a model vessel portion and a diameter of a model stem vesselproximal to the vessel crown from the model biological tree image,calculating patient resistance based upon at least the length of thepatient vessel portion and the diameter of the patient stem vesselproximal to the vessel crown from the patient's biological tree image,and comparing the calculated model resistance to the calculated patientresistance to determine the extent of vessel and/or organ disease. Inanother embodiment, the steps of calculating model resistance andcalculating patient resistance are further based upon the use of aconstant. In yet another embodiment, the patient resistance is aresistance of a stem segment, and wherein the length of the patientvessel portion is a length of a stem. In an additional embodiment, thestep of calculating the resistance of a stem segment is performed bydividing the length of a stem by the diameter of the patient stem vesselproximal to the vessel crown to the fourth power multiplied by aconstant for the stem, or a mathematical equivalent thereof. In yet anadditional embodiment, the constant for the stem is equivalent to onehundred and twenty eight multiplied by a fluid viscosity divided by pi,or a mathematical equivalent thereof.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the patient resistance is a resistance of acrown, and wherein the length of the patient vessel portion is a crownlength. In another embodiment, the step of calculating the resistance ofa crown is further based upon the use of a constant for the crown. Inyet another embodiment, the constant for the crown is equivalent to amaximum resistance multiplied by a maximum diameter to the fourth powerdivided by a maximum length, or a mathematical equivalent thereof. In anadditional embodiment, the step of calculating the resistance of a crownis performed by dividing the crown length by the diameter of the patientstem vessel proximal to the vessel crown to the fourth power multipliedby a constant, or a mathematical equivalent thereof. In yet anadditional embodiment, the patient resistance is a total resistance ofthe biological tree, and wherein the length of the patient vesselportion is a cumulative biological tree vessel length. In anotherembodiment, the step of calculating the total resistance of thebiological tree is further based upon the use of a constant for thecrown.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the step of calculating the total resistanceof the biological tree is performed by dividing the cumulativebiological tree vessel length by the diameter of the patient stem vesselproximal to the vessel crown to the fourth power multiplied by aconstant, or a mathematical equivalent thereof. In another embodiment,the patient resistance is the resistance of blood within a vesselportion. In yet another embodiment, the patient vessel portion comprisesa vessel selected from the group consisting of a blood vessel, a bileduct, a lung, and a neuron. In an additional embodiment, the patientresistance is the resistance of a gas within a vessel portion. In yet anadditional embodiment, the gas comprises breathable air, and wherein thevessel portion comprises a portion of a lung.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the step of comparing the calculated modelresistance to the calculated patient resistance is performed bygraphically comparing patient resistance data to model resistance datato determine the extent of vessel and/or organ disease by identifyingone or more graphical differences between said data. In anotherembodiment, the step of comparing calculated model resistance to thecalculated patient resistance is performed by comparing patientresistance data to model resistance data in table form to determine theextent of vessel and/or organ disease by identifying one or morenumerically calculated differences between said data. In yet anotherembodiment, the model biological tree is generated from a minimum energyhypothesis calculation.

In at least one embodiment of a method for determining vessel volume,the method comprises the steps of obtaining a biological tree imageshowing a structure of at least part of a biological tree, identifying adiameter of a stem segment and a length of a stem segment from thebiological tree image, and calculating cumulative crown volume basedupon at least the diameter of the stem segment and the length of thestem segment. In another embodiment, the step of calculating cumulativecrown volume is further based upon the use of a constant. In yet anotherembodiment, the step of calculating cumulative crown volume is performedby multiplying a constant by a product of the diameter of a stem segmentto the two-thirds power and the length of a stem segment, or amathematical equivalent thereof. In an additional embodiment, thediameter of a stem segment is a most proximal stem diameter, wherein thelength of a stem segment is a cumulative biological tree vessel length,and wherein the cumulative crown volume is a cumulative biological treevolume. In yet an additional embodiment, the step of calculating thecumulative biological tree volume is further based upon the use of aconstant.

In at least one embodiment of a method for determining vessel volume,the step of calculating cumulative biological tree volume is performedby multiplying a constant by a product of the most proximal stemdiameter to the two-thirds power and the cumulative biological treevessel length, or a mathematical equivalent thereof. In anotherembodiment, the method further comprises the step of comparing thecalculated cumulative crown volume to a calculated model cumulativecrown volume to determine the extent of vessel and/or organ disease. Inyet another embodiment, the step of comparing the calculated cumulativecrown volume to the calculated model cumulative crown volume isperformed by graphically comparing calculated cumulative crown volumedata to calculated model cumulative crown volume data to determine theextent of vessel and/or organ disease by identifying one or moregraphical differences between said data. In an additional embodiment,the step of comparing the calculated cumulative crown volume to thecalculated model cumulative crown volume is performed by comparingcalculated cumulative crown volume data to calculated model cumulativecrown volume data in table form to determine the extent of vessel and/ororgan disease by identifying one or more numerically calculateddifferences between said data. In yet an additional embodiment, themethod further comprises the steps of identifying a most proximal stemdiameter from the biological tree image, identifying a cumulativebiological tree vessel length from the biological tree image, andcalculating cumulative biological tree volume based upon at least themost proximal stem diameter and the cumulative biological tree vessellength.

In at least one embodiment of a method for determining vessel volume,the method further comprises the step of determining astructure-function scaling relation, wherein the structure-functionscaling relation comprises a first component and a second component, ora mathematical equivalent thereof. In another embodiment, the firstcomponent comprises the cumulative crown volume divided by thecumulative biological tree volume, and Wherein the second componentcomprises the diameter of a stem segment divided by the most proximalstem diameter, and wherein the first component equals the secondcomponent to the third power, or a mathematical equivalent thereof. Inyet another embodiment, the method further comprises the step ofdetermining a non-dimensional structure-function scaling relation,wherein the non-dimensional structure-function scaling relationcomprises a first component, a second component, and a third component,or a mathematical equivalent thereof. In an additional embodiment, thefirst component comprises the cumulative crown volume divided by thecumulative biological the volume, wherein the second component comprisesthe diameter of a stem segment divided by the most proximal stemdiameter, wherein the third component comprises the length of a stemsegment divided by the cumulative biological tree vessel length, andwherein the first component equals the second component to thetwo-thirds power multiplied by the third component, or a mathematicalequivalent thereof. In yet an additional embodiment, the calculatedcumulative crown volume provides information useful for a diagnosis of adisease.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the method comprising the steps of obtaininga model biological tree, obtaining a patient's biological tree imageshowing a vasculature of at least part of a patient's biological tree,identifying a diameter of a patient stem segment and a length of apatient stem segment from the patients biological tree image,calculating model cumulative crown volume based upon at least a diameterof a model stem segment and a length of a model stem segment from themodel biological tree, calculating patient cumulative crown volume basedupon at least the diameter of the patient stem segment and the length ofthe patient stem segment the patient's biological tree image, andcomparing the calculated model cumulative crown volume to the calculatedpatient cumulative crown volume to determine the extent of vessel and/ororgan disease. In another embodiment, the step of calculating patientemulative crown volume is further based upon the use of a constant. Inyet another embodiment, the step of calculating patient cumulative crownvolume is performed by multiplying a constant by a product of thediameter of the patient stem segment to the two-thirds power and thelength of the patient stem segment, or a mathematical equivalentthereof. In an additional embodiment, the diameter of the patient stemsegment is a most proximal stem diameter, wherein the length of thepatient stem segment is a cumulative biological tree vessel length, andwherein the patient cumulative crown volume is a cumulative biologicaltree volume. In yet an additional embodiment, the step of calculatingthe cumulative biological tree volume is further based upon the use of aconstant.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the step of calculating cumulative biologicaltree volume is performed by multiplying a constant by a product of themost proximal stem diameter to the two-thirds power and the cumulativebiological tree vessel length, or a mathematical equivalent thereof. Inanother embodiment, the step of comparing the calculated modelcumulative crown volume to the calculated model cumulative crown volumeis performed by graphically comparing calculated model cumulative crownvolume data to calculated model cumulative crown volume data todetermine the extent of vessel and/or organ disease by identifying oneor more graphical differences between said data. In yet anotherembodiment, the step of comparing the calculated model cumulative crownvolume to the calculated model cumulative crown volume is performed bycomparing calculated model cumulative crown volume data to calculatedmodel cumulative crown volume data in table form to determine the extentof vessel and/or organ disease by identifying one or more numericallycalculated differences between said data. In an additional embodiment,the method further comprises the steps of identifying a most proximalpatient stem diameter from the patient's biological tree image,identifying a patient's cumulative biological tree vessel length fromthe patient's biological tree image, and calculating a patient'scumulative biological tree volume of the patient's biological tree basedupon at least the most proximal patient stem diameter and the patient'scumulative biological tree vessel length. In yet an additionalembodiment, the method further comprises the step of determining astructure-function scaling relation, wherein the structure-functionscaling relation comprises a first component and a second component, ora mathematical equivalent thereof.

In at least one embodiment of a method for diagnosing disease in apatient's biological tree, the first component comprises the patient'scumulative crown volume divided by the patient's cumulative biologicaltree volume, and wherein the second component comprises the diameter ofthe patient stem segment divided by the most proximal patient stemdiameter, and wherein the first component equals the second component tothe third power, or a mathematical equivalent thereof. In anotherembodiment, the method further comprises the step of determining anon-dimensional structure-function scaling relation, wherein thenon-dimensional structure-function scaling relation comprises a firstcomponent, a second component, and a third component, or a mathematicalequivalent thereof. In yet another embodiment, the first componentcomprises the patient's cumulative crown volume divided by the patient'scumulative biological tree volume, wherein the second componentcomprises the diameter of the patient stem segment divided by the mostproximal patient stem diameter, wherein the third component comprisesthe length of the patient stem segment divided by the patient'scumulative biological tree vessel length, and wherein the firstcomponent equals the second component to the two-thirds power multipliedby the third component, or a mathematical equivalent thereof. In anadditional embodiment, the calculated patient's cumulative crown volumeprovides information useful for a diagnosis of a disease.

In at least one embodiment of a system for determining the resistance ofa flow within at least a portion of a vessel, the system comprises aprocessor, a storage medium operably connected to the processor, thestorage medium capable of receiving and storing data relative ofmeasurements from a vasculature of a vessel, wherein the processor isoperable to perform one or more steps of one or more of theaforementioned methods.

In at least one embodiment of a program having a plurality of programsteps to be executed on a computer having a processor and a storagemedium to analyze data relative of a vasculature of a vessel, theprogram is operable to perform one or more steps of one or more of theaforementioned methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the relation between normalized cumulative arterial volumeand corresponding normalized cumulative arterial length for each crownon a log-log plot, according to at least one embodiment of the presentdisclosure;

FIG. 2 shows the presence of DCAD at locations along the mean trendlines for normal (solid) and DCAD vasculature (broken) according to atleast one embodiment of the present disclosure;

FIG. 3 shows a diagnostic system according to at least one embodiment ofthe present disclosure;

FIG. 4 shows an illustration of a definition of a stem-crown unitaccording to at least one embodiment of the present disclosure;

FIGS. 5A-5C show relationships between resistance and diameter andnormalized crown length of LAD, LCx, and RCA trees of a pig,respectively, according to at least one embodiment of the presentdisclosure;

FIGS. 5D-5F show relationships between resistance and length of LAD,LCx, and RCA trees of a pig, respectively, according to at least oneembodiment of the present disclosure;

FIG. 6A shows a relationship between resistance and diameter andnormalized crown length in symmetric vascular trees for various species,according to at least one embodiment of the present disclosure;

FIG. 6B shows a relationship between resistance and length in symmetricvascular trees for various species, according to at least one embodimentof the present disclosure;

FIG. 7A shows a table of parameters with correlation coefficientscalculated from the Marquardt-Levenberg algorithm for various species,according to at least one embodiment of the present disclosure;

FIG. 7B shows a comparison of data from nonlinear regression andequations of the present disclosure; according to at least oneembodiment of the present disclosure;

FIG. 8A shows a relationship between resistance and diameter andnormalized crown length in the LAD, LCx, and RCA epicardial trees of apig, respectively, according to at least one embodiment of the presentdisclosure;

FIG. 8B shows a relationship between resistance and length in the LAD,LCx, and RCA epicardial trees of a pig, respectively, according to atleast one embodiment of the present disclosure;

FIG. 9 shows a table of parameters B and A in asymmetric coronary treesand corresponding: epicardial trees with vessel diameters greater than 1mm, according to at least one embodiment of the present disclosure;

FIG. 10 shows a table of parameters B and A in various organs, accordingat least one embodiment of the present disclosure;

FIG. 11 shows a table of parameter A obtained from nonlinear regressionin various organs, according to at least one embodiment of the presentdisclosure;

FIGS. 12A-12C show relations between diameter and length and normalizedcrown volume in the LAD, LCx, and RCA trees of a pig, respectively,according to at least one embodiment of the present disclosure;

FIG. 13 shows a relation between diameter and length and normalizedcrown volume in the LAD, LCx, and RCA epicardial trees of a pig,respectively, according to at least one embodiment of the presentdisclosure;

FIG. 14 shows a relation between diameter and length and normalizedcrown volume in the symmetric vascular tree for various organs andspecies, according to at least one embodiment of the present disclosure;and

FIG. 15 shows a comparison of data from nonlinear regression and anequation of the present disclosure; according to at least one embodimentof the present disclosure.

DETAILED DESCRIPTION

The disclosure of the present application applies concepts frombiomimetics and microfluidics to analyze vascular tree structure, thusimproving the efficacy and accuracy of diagnostics involving vasculardiseases such as DCAD. Scaling laws are developed in the form ofequations that use the relationships between arterial volume,cross-sectional area, blood flow and the distal arterial length toquantify moderate levels of diffuse coronary artery disease. For thepurposes of promoting an understanding of the principles of the presentdisclosure, reference will now be made to the embodiments illustrated inthe drawings, and specific language will be used to describe the same.It will nevertheless be understood that no limitation of the scope ofthe present disclosure is thereby intended.

Biomimeties (also known as bionics, biognosis, biomimicry, or bionicalcreativity engineering) is defined as the application of methods andsystems found in nature to the study and design of engineering systemsand modern technology. The mimic of technology from nature is based onthe premise that evolutionary pressure forces natural systems to becomehighly optimized and efficient. Some examples include (1) thedevelopment of dirt- and water-repellent paint from the observation thatthe surface of the lotus flower plant is practically unsticky, (2) hullsof boats imitating the thick skin of dolphins, and (3) sonar, radar, andmedical ultrasound imaging imitating the echolocation of bats.

Microfluidics is the study of the behavior, control and manipulation ofmicroliter and nanoliter volumes of fluids. It is a multidisciplinaryfield comprising physics, chemistry, engineering and biotechnology, withpractical applications to the design of systems in which such smallvolumes of fluids may be used. Microfluidics is used in the developmentof DNA chips, micro-propulsion, micro-thermal technologies, andlab-on-a-chip technology.

Regarding the minimum energy hypothesis, the architecture (or manifolds)of the transport network is essential for transport of material inmicrofluid channels for various chips. The issue is how to design newdevices, and more particularly, how to fabricate microfluidic channelsthat provide a minimum cost of operation. Nature has developed optimalchannels (or transport systems) that utilize minimum energy fortransport of fluids. The utility of nature's design of transport systemsin engineering applications is an important area of biomimetics.

Biological trees (for example, vascular trees) are either used toconduct fluids such as blood, air, bile or urine. Energy expenditure isrequired for the conduction of fluid through a tree structure because offrictional losses. The frictional losses are reduced when the vesselbranches have larger diameters. However, this comes with a costassociated with the metabolic construction and maintenance of the largervolume of the structure. The question is what physical or physiologicalfactors dictate the design of vascular trees. The answer is that thedesign of vascular trees obeys the “minimum energy hypothesis”, i.e.,the cost of construction and operation of the vascular system appears tobe optimized.

The disclosure of the present application is based on a set of scalinglaws determined from a developed minimum energy hypothesis. Equation #1(the “volume-length relation”) demonstrates a relationship betweenvessel volume, the volume of the entire crown, vessel length, and thecumulative vessel length of the crown:

$\begin{matrix}{\frac{V}{V_{\max}} = ( \frac{L}{L_{\max}} )^{\frac{5}{ɛ^{\prime} + 1}}} & (1)\end{matrix}$

In Equation #1, V represents the vessel volume, V_(max) the volume ofthe entire crown, L represents the vessel length, L_(max) represents thecumulative vessel length of the entire crown, and ε^(t) represents thecrown flow resistance, which is equal to the ratio of metabolic toviscous power dissipation.

Equation 42 (the “diameter-length relation”) demonstrates a relationshipbetween vessel diameter, the diameter of the most proximal stem, vessellength, and the cumulative vessel length of the crown:

$\begin{matrix}{\frac{D}{D_{\max}} = ( \frac{L}{L_{\max}} )^{\frac{{3ɛ^{\prime}} - 2}{4{({ɛ^{\prime} + 1})}}}} & (2)\end{matrix}$

In Equation #2, D represents the vessel diameter, D_(max) represents thediameter of the most proximal stem, L represents the vessel length,L_(max) represents the cumulative vessel length of the entire crown, andε^(t) represents the crown flow resistance, which is equal to the ratioof metabolic to viscous power dissipation.

Equation #3 (the “flow rate-diameter relation”) demonstrates arelationship between the flow rate of a stem, the flow rate of the mostproximal stem, vessel diameter, and the diameter of the most proximalstem:

$\begin{matrix}{\frac{Q}{Q_{\max}} = ( \frac{D}{D_{\max}} )^{\frac{4{({ɛ^{\prime} + 1})}}{{3ɛ^{\prime}} - 2}}} & (3)\end{matrix}$

In Equation Q represents flow rate of a stem, Q_(max) represents theflow rate of the most proximal stem, V represents vessel diameter,V_(max) represents the diameter of the most proximal stem, and ε^(t)represents the crown flow resistance, which is equal to the ratio ofmetabolic to viscous power dissipation.

Regarding the aforementioned Equations, a vessel segment is referred toas a “stem,” and the entire tree distal to the stern is referred as a“crown.” The aforementioned parameters relate to the crown flowresistance and is equal to the ratio of maximum metabolic-to-viscouspower dissipation.

Two additional relations were found for the vascular trees. Equation #4(the “resistance-length and volume relation”) demonstrates arelationship between the crown resistance, the resistance of the entiretree, vessel length, the cumulative vessel length of the crown, vesselvolume, and the volume of the entire crown:

$\begin{matrix}{\frac{R_{c}}{R_{\max}} = \frac{( {L/L_{\max}} )^{3}}{( {V/V_{\max}} )^{ɛ^{\prime}}}} & (4)\end{matrix}$

In Equation #4, R_(c) represents the crown resistance, R_(max)represents the resistance of the entire tree, L represents vessellength, L_(max) represents the cumulative vessel length of the entirecrown, V represents vessel volume, V_(max) represents the volume of theentire crown, and ε^(t) represents the crown flow resistance, which isequal to the ratio of metabolic to viscous power dissipation.Resistance, as referenced herein, is defined as the ratio of pressuredifferenced between inlet and outlet of the vessel.

Equation #5 (the “flow rate-length relation”) demonstrates arelationship between the flow rate of a stem, the flow rate of the mostproximal stem, vessel length, the cumulative vessel length of the entirecrown:

$\begin{matrix}{\frac{Q}{Q_{\max}} = \frac{L}{L_{\max}}} & (5)\end{matrix}$

In Equation #5, Q represents flow rate of a stem, Q_(max) represents theflow rate of the most proximal stem, L represents vessel length, andL_(max) represents the cumulative vessel length of the entire crown.

In at least one embodiment of the disclosure of the present application,the application of one or more of the aforementioned Equations toacquired vessel data may be useful diagnose and/or and in the diagnosisof disease.

By way of example, the application of one or more of the aforementionedEquations are useful to diagnose DCAD. For such a diagnosis, theapplications of Equations #1-#3 may provide the “signatures” of normalvascular trees and impart a rationale for diagnosis of diseaseprocesses. The self-similar nature of these laws implies that theanalysis can be carried out on a partial tree as obtained from anangiogram, a computed tomography (CT) scan, or an magnetic resonanceimaging (MRI). Hence, the application of these Equations to the obtainedimages may serve for diagnosis of vascular disease that affect the lumendimension, volume, length (vascularity) or perfusion (flow rate).Additionally, the fabrication of the microfluidic channels can begoverned by Equations #1-#5 to yield a system that requires minimumenergy of construction and operation. Hence, energy requirements will beat a minimum to transport the required microfluidics.

In one exemplary embodiment, the application of the volume-lengthrelation (Equation #1) to actual obtained images is considered as shownin FIG. 1. First, images (angiograms in this example) of swine coronaryarties were Obtained. The application of Equation #1 on various volumesand lengths from the angiograms resulted in the individual data pointsshown within FIG. 1 (on a logarithmic scale). The line depicted withinFIG. 1 represents the mean of the data points (the best fit) among theidentified data points.

In FIG. 2, the mean of the data (solid line) is compared to an animalwith diffuse disease at three different vessel sizes: proximal (1),middle (2), and distal (3). The reductions in volume shown on FIG. 2correspond to approximately 40% stenosis, which is typicallyundetectable with current methodologies. At each diffuse stenosis, thelength remains constant but the diameter (cross-sectional, and hence,volume) changes. The length is unlikely to change unless the flowbecomes limiting (more than approximately 80% stenosis) and the vascularsystem experiences vessel loss (rarefication) and remodeling. It isclear that a 40% stenosis deviates significantly from the y-axis (asdetermined by statistical tests) from the normal vasculature, and assuch, 40% stenosis can be diagnosed by the system and method of thedisclosure of the present application. It can he appreciated that thedisclosure of the present application can predict inefficiencies as lowas about 10%, compared to well-trained clinicians who can only predictinefficiencies at about 60% at best.

This exemplary statistical test compares the deviation of disease tonormality relative to the variation within normality. The location ofthe deviation along the x-axis corresponds to the size of the vessel.The vessel dimensions range as proximal=mid>distal. Hence, by utilizingthe system and method of the disclosure of the present application, thediagnosis of the extent of disease and the dimension of the vesselbranch is now possible. Similar embodiments with other scaling relationsas described herein can be applied similarly to model and actualvascular data.

The techniques disclosed herein have tremendous application in a largenumber of technologies. For example, a software program or hardwaredevice may be developed to diagnose the percentage of inefficiency(hence, occlusion) in a circulatory vessel or system.

Regarding the computer-assisted determination of such diagnoses, anexemplary system of the disclosure of the present application isprovided. Referring now to FIG. 3, there is shown a diagrammatic view ofan embodiment of diagnostic system 300 of the present disclosure. In theembodiment shown in FIG. 3, diagnostic system 300 comprises user system302. In this exemplary embodiment, user system 302 comprises processor304 and one or more storage media 306. Processor 304 operates upon dataobtained by or contained within user system 302. Storage medium 306 maycontain database 308, whereby database 308 is capable of storing andretrieving data. Storage media 306 may contain a program (including, butnot limited to, database 308), the program operable by processor 304 toperform a series of steps regarding data relative of vessel measurementsas described in further detail herein.

Any number of storage media 306 may be used with diagnostic system 300of the present disclosure, including, but not limited to, one or more ofrandom access memory, read only memory. EPROMs, hard disk drives, floppydisk drives, optical disk drives, cartridge media, and smart cards, forexample. As related to user system 302, storage media 306 may operate bystoring data relative of vessel measurements for access by a user and/orfor storing computer instructions. Processor 304 may also operate upondata stored within database 308.

Regardless of the embodiment of diagnostic system 300 referenced hereinand/or contemplated to be within the scope of the present disclosure,each user system 302 may he of various configurations well known in theart. By way of example, user system 302, as shown in FIG. 3, compriseskeyboard 310, monitor 312, and printer 314. Processor 304 may furtheroperate to manage input and output from keyboard 310, monitor 312, andprinter 314. Keyboard 310 is an exemplary input device, operating as ameans for a user to input information to user system 302. Monitor 312operates as a visual display means to display the data relative ofvessel measurements and related information to a user using a usersystem 302. Printer 314 operates as a means to display data relative ofvessel measurements and related information. Other input and outputdevices, such as a keypad, a computer mouse, a fingerprint reader, apointing device, a microphone, and one or more loudspeakers arecontemplated to be within the scope of the present disclosure. It can beappreciated that processor 304, keyboard 310, monitor 312, printer 314and other input and output devices referenced herein may be componentsof one or more user systems 302 of the present disclosure.

It can be appreciated that diagnostic system 300 may further compriseone or more server systems 316 in bidirectional communication with usersystem 302, either by direct communication (shown by the single lineconnection on FIG. 3), or through a network 318 (shown by the doubleline connections on FIG. 3) by one of several configurations known inthe art. Such server systems 316 may comprise one or more of thefeatures of a user system 302 as described herein, including, but notlimited to, processor 304, storage media 306, database 308, keyboard310, monitor 312, and printer 314, as shown in the embodiment ofdiagnostic system 300 shown in FIG. 3. Such server systems 316 may allowbidirectional communication with one or more user systems 302 to allowuser system 302 to access data relative of vessel measurements andrelated information from the server systems 316. It can be appreciatedthat a user system 302 and/or a server system 316 referenced herein maybe generally referred to as a “computer.”

Several concepts are defined to formulate resistance scaling laws of thedisclosure of the present application. A vessel segment is defined as a“stem” and the entire tree distal to the stem is defined as a “crown,”as shown in FIG. 4 and as previously disclosed herein. FIG. 4 shows aschematic illustration of the definition of the stem-crown unit. Threestem-crown units are shown successively (1, 2, and n), with the smallestunit corresponding to an arteriole-capillary or venule-capillary unit.An entire vascular tree, or substantially the entire vascular tree,consists of many stem-crown units down to, for example, the smallestarterioles or venules. In one exemplary embodiment of the disclosure ofthe present application, the capillary network (referenced herein ashaving vessel diameters of less than 8 microns) is excluded from theanalysis because it is not tree-like in structure. A stem, for purposesof simplification, is assumed to be a cylindrical tube with noconsideration of vessel tapering and other nonlinear effects as theyplay a relatively minor role in determining the hemodynamics of theentire tree. However, the disclosure of the present application is notintended to be limited by the aforementioned capillary network exclusionand/or the aforementioned stem assumption.

Through the Hagen-Poiseuille law known in the art, the resistance of thesteady laminar flow in a stem of an entire tree may be provided as shownin Equation #6:

$\begin{matrix}{R_{s} = \frac{\Delta \; P_{s}}{Q_{s}}} & (6)\end{matrix}$

In Equation #6, R_(s) is the resistance of a stem segment, ΔP_(s) is thepressure gradient along the stem, and Q_(s) is a volumetric flow ratethrough the stem.

According to the disclosure of the present application, Equation #6,providing R_(s), may be written in a form considering stem length anddiameter, as shown in Equation #7.

$\begin{matrix}{R_{s} = {\frac{128\mu \; L_{s}}{\pi \; D_{s}^{4}} = {K_{s}\frac{L_{s}}{D_{s}^{4}}}}} & (7)\end{matrix}$

In Equation #7, R_(s) is the resistance of a stem segment, L_(s) is thelength of the stem, D_(s) is the diameter of the stem, μ is theviscosity of a fluid, and K_(s) is a constant equivalent to 128 μ/π.

Furthermore, the resistance of a crown may be demonstrated as shown inEquation #8:

$\begin{matrix}{R_{s} = \frac{\Delta \; P_{c}}{Q_{s}}} & (8)\end{matrix}$

In Equation #8, R_(c), is the crown resistance, ΔP_(s) is the pressuregradient in the crown from the stem to the terminal vessels, and Q_(s)is a volumetric flow rate through the stem. Equation #8 may also bewritten in a novel form to solve for R_(c) in accordance with thedisclosure of the present application as shown in Equation #9:

$\begin{matrix}{R_{c} = {K_{c}\frac{L_{c}}{D_{s}^{4}}}} & (9)\end{matrix}$

In Equation #9, R_(c) is the crown resistance, L_(c) is the crownlength, D_(s) is the diameter of the stem vessel proximal to the crown,and K_(c) is a constant that depends on the branching ration, diameterratio, the total number of tree generations, and viscosity in the crown.The crown length, L_(c), may be defined as the sum of the lengths ofeach vessel in the crown (or substantially all of the vessels in thecrown).

As Equation #9, according to the disclosure of the present application,is applicable to any stem-crown unit, one may obtain the followingequation:

$\begin{matrix}{R_{\max} = {K_{c}\frac{L_{\max}}{D_{\max}^{4}}}} & (10)\end{matrix}$

so that the following formula for K_(c) may be obtained:

$\begin{matrix}{K_{c} = \frac{R_{\max} \cdot D_{\max}^{4}}{L_{\max}}} & (11)\end{matrix}$

D_(max), L_(max), and R_(max) correspond to the most proximal sterndiameter, the cumulative vascular length, and total resistance of theentire tree, respectively. In the non-dimensional form. Equation #11 canhe written as:

$\begin{matrix}{{( \frac{R_{c}}{R_{\max}} ) \cdot ( \frac{D_{s}}{D_{\max}} )^{4}} = {A_{1}( \frac{L_{c}}{L_{\max}} )}} & (12)\end{matrix}$

Parameter A₁ in Equation #12, as provided above, should be equal to one.From Equations #7 and #9, one may then obtain the desired resistancescaling relation between a single vessel (a stern) and the distal crowntree:

$\begin{matrix}{( \frac{R_{s}}{R_{c}} ) = {\frac{K_{s}}{K_{c}}( \frac{L_{s}}{L_{c}} )}} & (13)\end{matrix}$

Equations #7-13 relate the resistance of a single vessel to thecorresponding distal tree.

Verification. The asymmetric coronary arterial trees of hearts andsymmetric vascular trees of many organs were used to verify the proposedresistance scaling law. First, the asymmetric coronary arterial tree hasbeen reconstructed in pig hearts by using the growth algorithmintroduced by Mittal et d (A computer reconstruction of the entirecoronary arterial tree based on detailed morphometric data. Ann. Biomed.Eng. 33 (8):1015-1026 (2005)) based on measured morphometric data ofKassab et al. (Morphometry of pig coronary arterial trees. Am J PhysiolHeart Circ Physiol. 265:H350-H365 (1993)). Briefly, vessels ≧40 μm werereconstructed from cast data while vessels <40 μm were reconstructedfrom histological data. After the tree was reconstructed, each vesselwas assigned by diameter-defined Strahler orders which was developedbased on the Strahler system (Strahler, A. N. Hypsometric (areaaltitude) analysis of erosional topology. Bull Geol Soc Am. 63:1117-1142(1952)).

Furthermore, symmetric vascular trees of many organs were constructed inthe Strahler system, based on the available literature. Here, thepulmonary arterial tree of rats was obtained from the study of Jiang etal. (Diameter-defined Strahler system and connectivity matrix of thepulmonary arterial tree. J. Appl. Physiol. 76:882-892 (1994)); thepulmonary arterial/venous trees of cats from Yen et al. (Morphometry ofcat's pulmonary arterial tree. J. Biomech. Eng. 106:131-136 (1984) andMorphometry of cat pulmonary venous tree. J Appl. Physiol. Respir.Environ. Exercise. Physiol. 55:236-242 (1983)); the pulmonary arterialtrees of humans from Singhal et al. (Morphometric study of pulmonaryarterial tree and its hemodynamics. J. Assoc. Physicians India.21:719-722 (1973) and Morphometry of the human pulmonary arterial tree.Circ. Res. 33:190 (1973)) and Huang et (Morphometry of the humanpulmonary vasculature. J. Appl. Physiol. 81:2123-2133 (1996)); thepulmonary venous trees of humans from Horsfield et al. (Morphometry ofpulmonary veins in man. Lung 159:211-218 (1981)) and Huang et al.; theskin muscle arterial tree of hamsters from Bertuglia et al. (Hypoxia- orhyperoxia-induced changes in arteriolar vasomotion in skeletal musclemicrocirculation. Am J Physiol Heart Circ Physiol. 260:H362-H372(1991)); the retractor muscle arterial tree of hamsters from Ellsworthet al. (Analysis of vascular pattern and dimensions in arteriolarnetworks of the retractor muscle in young hamsters. Microvase. Res.34:168-183 (1987)); the mesentery arterial tree of rats from Ley et al.(Topological structure of rat mesenteric microvessel networks.Microvasc. Res. 32:315-332 (1986)); the sartorius muscle arterial treeof cats from Koller et al. (Quantitative analysis of arteriolar networkarchitecture in cat sartorius muscle. Am J Physiol Heart Circ Physiol.253: H154-H164 (1987)); and the bulbular conjunctiva arterial/venoustrees of humans and the omentum arterial tree of rabbits from Fenton etal. (Microcirculatory model relating geometrical variation to changes inpressure and flow rate. Ann. Biomed Eng. 1981; 9:303-321 (1981)).

Data analysis. For the asymmetric coronary arterial trees, full treedata are presented as log-log density plots showing the frequency ofdata because of the enormity of data points, i.e., darkest shadereflects highest frequency or density and the lightest shade reflectsthe lowest frequency. The nonlinear regression (SigmaStat 3.5) is usedto analyze the data in both asymmetric and symmetric tree, which usesthe Marquardt-Levenberg algorithm (nonlinear regression) to find thecoefficients (parameters) of the independent variables that give the“best fit” between the equation and the data.

Results: Validation of resistance scaling law in entire vascular trees.The predictions of these novel scaling laws were then validated in boththe asymmetric coronary trees and the symmetric vascular trees for whichthere exists morphometric data in the literature (e.g., vessels ofvarious skeletal muscles, mesentery, omentum, and conjunctiva).

First, the entire asymmetric coronary LAD, LCx, and RCA trees withseveral millions of vessels were analyzed(15,16). FIGS. 5A, 5B, and 5Cshow a log-log plot of (R_(c)/R_(max))·(D_(s)/D_(max))⁴ as a function ofnormalized crown length (L_(c)/L_(max)) for LAD, LCx, and RCA trees,respectively. Relationships between (R_(c)/R_(max))·(D_(s)/D_(max))⁴ andnormalized crown length (L_(c)/L_(max)) in the asymmetric entire LAD(FIG. 5A), LCx (FIG. 5B), and RCA (FIG. 5C) trees of pig, which include946937, 571383, and 836712 stem-crown units are shown, respectively.Through the Marquardt-Levenberg algorithm with the exponents ofL_(c)/L_(max) constrained to one, parameter A₁ in Equation #12 has avalue of 1.027 (R²=0.990), 0.993 (R²=0.997), and 1.084 (R²=0.975) forLAD, LCx, and RCA trees, respectively. The values of A₁ obtained frommorphometric data are in agreement with the theoretical value of one.Corresponding to FIGS. 5A, 5B, and 5C, FIGS. 5D, 5E, and 5F show alog-log plot of R_(c)/R_(s) as a function of L_(c)/L_(s). ParameterK_(s)/K_(c) in Equation #13 has a value of 2.647 (R²=0.954), 2.943(R²=0.918), and 2.147 (R²=0.909) for LAD, LCx, and RCA trees,respectively. FIGS. 5D, 5E, and 5F show a relationship betweenR_(c)/R_(s) and L_(c)/L_(s) in the LAD, LCx, and RCA trees of pig,corresponding to FIGS. 5A, 5B, and 5C.

Furthermore, FIGS. 6A and 6B show the log-log plots of(R_(c)/R_(max))·(D₀/D_(max))⁴ and R_(c)/R_(s) as a function ofL_(c)/L_(max) and L_(c)/L_(s), respectively, in the vascular trees ofvarious species. Corresponding to FIGS. 6A and 6B, theMarquardt-Levenherg algorithm was used to calculate the parameters A₁and K_(s)/K_(c) in Equations #12 and #13, respectively, while theexponents of L_(c)/L_(max) and L_(c)/L_(s) were constrained to be one.Parameters A₁ in Equation #12 and K_(s)/K_(c) in Equation #13 withcorrelation coefficient for various species are listed in the tableshown in FIG. 7A. The data in FIG. 7A have a mean value (averaged overall organs and species) of 1.01±0.06 for parameter A₁. FIG. 7B shows acomparison of (K_(s)/K_(c))_(ML) from the nonlinear regression ofanatomical data and (K_(s)/K_(c))_(EQ) based on Equations K_(s)=128μ/πand

${K_{c} = \frac{R_{\max} \cdot D_{\max}^{4}}{L_{\max}}},$

noting that the comparison can be represented as

$( \frac{K_{s}}{K_{c}} )_{EQ} = {A \cdot {( \frac{K_{s}}{K_{c}} )_{ML}^{B}.}}$

When A is constrained to be one in the Marquardt-Levenberg algorithm, Bhas a value of one (R²=0.983). Using the same Marquardt-Levenbergalgorithm, a nonlinear regression fit of all raw data yields a mean of1.01 (R²=0.95) for parameter A₁. Both the mean value and the nonlinearregression fit of all data agree with the theoretical value of one.

FIG. 6B shows much smaller R_(c)/R_(s) in pulmonary vascular tree thanother organs at the same value of L_(c)/L_(s). Accordingly, theK_(s)/K_(c) values (shown in the table in FIG. 7A) are similar exceptfor the pulmonary vasculature with a larger value. The K_(s)/K_(c)values are also calculated based on Equations K_(s)=128μ/π andK_(c)=R_(max)·D_(max) ⁴/L_(max), which is compared with the K_(s)/K_(c)values obtained from the Marquardt-Levenberg algorithm, as shown in FIG.7B. The viscosity is determined based on an empirical in vivo relationthat depends on the vessel diameter. The comparison shows goodagreement. The K_(s)/K_(c) values in the pulmonary vasculature have alarger value because the cross-section area of pulmonary tree has alarge increase from proximal to terminal vessels in the pulmonary treeand the resistance of the entire tree (R_(max)) is much smaller. Theagreement between experimental measurement and theoretical relationsillustrate that the novel resistance scaling law disclosed herein ofEquations #9, #12, and #13 can be applied to a general vascular treedown to the smallest arterioles or venules.

Results: Resistance scaling law of partial vascular trees. FIGS. 8A and8B show the relations between (R_(c)/R_(max))·(D_(s)/D_(max))⁴ andnormalized crown volume (L_(c)/L_(max)) between R_(c)/R_(s) andL_(c)/L_(s), respectively, in the LAD, LCx, and RCA epicardial trees.FIG. 8A shows a relationship between (R_(c)/R_(max))·(D_(s)/D_(max))⁴and normalized crown volume (L_(c)/L_(max)) in the LAD, LCx, and RCAepicardial trees of pig with diameter of mother vessels larger than 1mm, which include 132, 90, and 192 vessel segments, respectively. FIG.8B shows a relationship between R_(c)/R_(s) and L_(c)/L_(s) in the LAD,LCx, and RCA epicardial trees of pig corresponding to FIG. 8A. ParameterA₁ in Equation #12 has a value of 0.902 (R²=0.907), 0.895 (R²=0.887),and 1.000 (R²=0.888) and parameter K_(s)/K_(c) in Equation #13 has avalue of 3.29 (R²=0.875), 148 (R²=0.816), and 3.12 (R²=0.927) for theLAD, LCx, and RCA epicardial trees, respectively.

The aforementioned study validates the novel resistance scaling law ofthe present disclosure that relates the resistance of a vessel branch tothe equivalent resistance of the corresponding distal tree in variousvascular trees of different organs and species. The significance of theresistant scaling law is that the hydraulic resistance of a distalvascular tree can be estimated from the proximal vessel segment. As aresult, the disclosure of the present application has wide implicationsfrom understanding fundamental vascular design to diagnosis of diseasein the vascular system.

Resistance scaling law. The mechanisms responsible for blood flowregulation in vascular trees are of central importance, but are stillpoorly understood. The arteriolar beds are the major site of vascularresistance, which contributes to the maintenance and regulation ofregional blood flow. Although arteriolar resistance plays an importantrole in the etiology of many diseases, in particular, hypertension, ithas been difficult to predict the resistance in the arteriolar beds. Thenovel resistance scaling law of the present disclosure addresses thisissue.

The resistance scaling laws (Equations #9, #12, and #13) are derivedbased on the relation of diameter ratio (DR=D_(i)/D_(i−1)), length ratio(LR=L_(i)/L_(i−1)) and branching ratio (BR=N_(i)/N_(i−1)) in a symmetrictree as:

${{DR} = {{{BR}^{- \frac{1}{2 + ɛ}}\mspace{14mu} {and}\mspace{14mu} {LR}} = {BR}^{- \frac{1}{3}}}},$

where ε=0 and ε=1 represent the area-preservation, πD_(i−1) ²=BR·πD_(i)², and Murray's law, πD_(i−1) ³=BR·πD_(i) ³, respectively.

Although the total cross-sectional area (CSA) may increase dramaticallyfrom the aorta to the arterioles, the variation is significantly smallerin most organs except for the lung. The increase of CSA towards thecapillaries is typically inferred from the decrease in velocity. Thevelocity between the most proximal and distal levels in various organsof mammals is found to vary by about a factor of five, except for thepulmonary vascular trees. This is clearly reflected by the table shownin FIG. 7A, in which

${K_{s}/K_{c}} = \frac{1}{K_{ɛ}}$

is relatively small except for the pulmonary vasculature. This impliesthat wall shear stress (WSS) increases from the arteries to thearterioles in most organs, which is consistent with previousmeasurements.

Structure-function scaling laws obtained from resistance scaling law. Amathematical model (the ¾-power scaling law) was derived in a symmetricvasculature to characterize the allometric scaling laws, based on theminimum energy theory. The ¾-power scaling law can be written asQ_(s)∝M^(3/4), where Q_(s) is the volumetric flow rate of the aorta andM is body mass. In a stem-crown unit, Q_(s) is the volumetric flow rateof the stem and M is the mass perfused by the stem crown unit. Thevolumetric flow rate of the stem is Q_(s)=πD_(s) ²U_(s)/4, where D_(s)and U_(s) are the diameter and the mean flow velocity of the stem(averaged over the cross-section of stem). Similar to at least one knownmodel, the pressure drop from the stein to the capillaries (ΔP_(c)) andthe mean flow velocity of the stem (U_(s)) are independent of theperfused mass so that D_(s)∝M^(3/8) and the resistance of the crown(R_(c)=ΔP_(c)/Q_(s)) is inversely proportional to the volumetric flowrate (R_(c)∝Q_(s) ⁻¹∝M^(−3/4)). Since D_(s)∝M^(3/8), R_(c)∝M^(−3/4), andK_(c) is a constant, Equations #9 and #12 yields that the crown lengthL_(c)∝M^(3/4). The cumulative length-mass scaling in pig hearts,L_(c)∝M^(3/4), has recently been verified by the present inventors andtheir research group. This relation, in conjunction with the flow-massrelation (Q_(s)∝M^(3/4)), yields the flow-length relation (Q_(s)∝L_(c))in the stem-crown unit, which has been previously validated.

Here, the crown length L_(c)∝M^(3/4) is different from the biologicallength l∝M^(1/4). The biological length (l) is the cumulative lengthalong a path from inlet (level zero) to the terminal (level N), but thecrown length is the total length of all vessels from inlet to theterminals. Although the biological length shows that the vascularphysiology and anatomy are four-dimensional, the crown length depicts a¾-power relation between the total length of entire/partial biologicalsystem and the perfused mass.

Clinical implications of resistance scaling law: The self-similar natureof the structure-function scaling laws in Equations #9, #12 and #13implies that they can be applied to a partial tree clinically (e.g., apartial tree obtained from an angiogram, computerized tomography, ormagnetic resonance imaging). As provided herein, the hypothesis usingthe LAD, LCx, and RCA epicardial pig trees obtained from casts truncatedat 1 mm diameter to mimic the resolution of noninvasive imagingtechniques was verified. The good agreement between experiments andtheory, as shown in FIG. 8, illustrates that the resistance scaling lawscan be applied to partial vascular trees as well as entire trees.

Significance of resistance scaling law: The novel resistance scaling law(Equations #9 and #12) provides a theoretical and physical basis forunderstanding the hemodynamic resistance of the entire tree (or asubtree) as well as to provide a rational for clinical diagnosis. Thescaling law illustrates the relationship between the structure (tree)and function (resistance), in which the crown resistance is proportionalto the crown length and inversely proportional to the fourth power ofstem diameter D_(s) ⁴. The small crown resistance corresponds to a smallcrown length, thus matching the transport efficiency of the crown. Anincrease of stem diameter can decrease the resistance, Which maycontribute to the self scaling of biological transport system. The novelscaling law provides an integration between a single unit and the whole(millions of units) and imparts a rationale for diagnosis of diseaseprocesses as well as assessment of therapeutic trials.

The disclosure of the present application provides a novel volumescaling law in a vessel segment and its corresponding distal tree ofnormal organs and in various species as, for example, V_(c)=K_(v)D_(s)^(2/3)L_(c), where V_(c) and L_(c) are the vascular volume and length,respectively, D_(s) is the diameter of vessel segment, and K_(v) is aconstant. A novel scaling relation of the disclosure of the presentapplication is validated with available vascular morphometric tree data,and may serve as a control reference to examine the change of bloodvolume in various organs under different states using conventionalimaging. A novel scaling law of the disclosure of the presentapplication is further validated through diameter-length, volume-length,flow-diameter, and volume-diameter scaling relations, derived based on aminimum energy hypothesis (15). Hence, the novel volume scaling law ofthe disclosure of the present application is consistent with a (minimumenergy) state of efficient vascular system.

In addition to the foregoing, it is known that V_(c)∝M (M is the massperfused by the stem-crown unit) from the ¾ allometric scaling law,where V_(c) is the crown volume (i.e., the sum of all vessel volumes inthe crown). Therefore, V_(c) can be represented as follows:

V _(c) =C _(v) M ^(1/4) M ^(3/4)   (14)

where C_(v) is a volume-mass constant.

There are two scaling relations: stem diameter-mass relation,D_(s)∝M^(3/8), wherein D_(s) is the diameter of stem vessel, and crownlength-mass relation, L_(c)∝M^(3/4), wherein L_(c) is the crown lengththat is defined as the sum of the lengths or substantially all of thelengths of each vessel in the crown).

From D_(s)=C_(d)M^(3/8), L_(c)=C_(l)M^(3/4), and Equation #14, one mayobtain:

$\begin{matrix}{V_{c} = {{C_{v}M^{1/4}M^{3/4}} = {{{C_{v}( \frac{D_{s}}{C_{d}} )}^{2/3}\frac{L_{c}}{C_{l}}} = {K_{v}D_{s}^{2/3}L_{c}}}}} & (15)\end{matrix}$

where K_(v)=C_(v)/(C_(d) ^(2/3)C_(l)) is a constant. Since Equation #15is applicable to any stem-crown unit, one may obtain

V_(max) = K_(v)D_(max)^(2/3)L_(max),

so that

${K_{v} = \frac{V_{\max}}{D_{\max}^{2/3}L_{\max}}},$

where D_(max), L_(max), and V_(max) correspond to the most proximal stemdiameter, the cumulative vascular length of entire tree, and thecumulative vascular volume of entire tree, respectively. Equation #15can also be made non-dimensional as:

$\begin{matrix}{( \frac{V_{c}}{V_{\max}} ) = {( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )}} & (16)\end{matrix}$

Morphometry of Vascular Trees. The volume scaling law of the disclosureof the present application is validated in the asymmetric entirecoronary arterial tree reconstructed in pig hearts through the growthalgorithm based on measured morphometric data. Furthermore, theasymmetric epicardial coronary arterial trees with vessel diametergreater than 1 mm were used to validate the scaling laws in partialvascular trees to mimic the resolution of medical imaging.

Symmetric vascular trees of many organs down to the smallest arterioleswere used to verify the proposed structure-function scaling law, whichwere constructed in the Strahler system, based on the availableliterature. The arterial and/or venous trees from the various specieswere obtained as previously referenced herein.

Data Analysis. All scaling relations (i.e., Equations #16 and #29-32)can be represented by a form of the type:

Y=A·X ^(B)   (17)

where X and Y are defined such that A and B should have theoreticalvalues of unity for Equation #16. X and Y are defined as

${( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )\mspace{14mu} {and}\mspace{14mu} ( \frac{V_{c}}{V_{\max}} )},$

respectively. For Equations #29-32, X and Y are defined as

${( \frac{L_{c}}{L_{\max}} )\mspace{14mu} {and}\mspace{14mu} ( \frac{D_{s}}{D_{\max}} )};{( \frac{L_{c}}{L_{\max}} )\mspace{14mu} {and}\mspace{14mu} ( \frac{V_{c}}{V_{\max}} )};$

and

${( \frac{D_{s}}{D_{\max}} )\mspace{14mu} {and}\mspace{14mu} ( \frac{Q_{s}}{Q_{\max}} )};{( \frac{D_{s}}{D_{\max}} )\mspace{14mu} {and}\mspace{14mu} ( \frac{V_{c}}{V_{\max}} )};$

respectively.

A nonlinear regression was then used to calculate A with B constrainedto 3/7, 1 2/7, 2⅓, and 3 for Equations #29-32, respectively. Thenonlinear regression uses the Marquardt-Levenberg algorithm to find theparameter, A, for the variables X and Y to provide the “best fit”between the equation and the data. In Equations #16 and #29-32, theparameter A should have a theoretical value of one.

Results.

Asymmetric Tree Model. The disclosure of the present applicationprovides a novel volume scaling law that relates the crown volume to thestem diameter and crown length in Equations #15 and #16. The validity ofEquations #15 and #16 were examined in the asymmetric entire (down tothe pre-capillary vessel segments) and epicardial (vessel diameter≧1 mm)LAD, LCx, and RCA trees of pig, as shown in FIGS. 12 and 13,respectively. FIG. 12 shows a relation between

$( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )$

and normalized crown volume in the entire asymmetric (a) LAD, (b) LCx,and (c) RCA trees of pig, which include 946,937, 571,383, and 836,712vessel segments, respectively. The entire tree data are presented aslog-log density plots showing the frequency of data because of theenormity of data points, i.e., darkest shade reflects highest frequencyor density and the lightest shade reflects the lowest frequency. FIG. 13shows a relation between

$( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )$

and normalized crown volume in the asymmetric LAD, LCx, and RCAepicardial trees of pig with vessel diameter larger than 1 mm, whichinclude 66, 42, and 71 vessel segments, respectively.

As shown in FIG. 9, exponent B is determined from a least-square fit,and parameter A is calculated by the nonlinear regression with theexponent B constrained to one. Both B and A for the entire asymmetricand partial trees show agreement with the theoretical value of one. Forthe table shown in FIG. 9, Parameters B (obtained from least-squarefits) and A (obtained from nonlinear regression with B constrained toone) in the asymmetric entire coronary trees and in the correspondingepicardial trees with vessel diameter>1 mm when Equation #16 isrepresented by Equation #17, where independent variables

${X = {{( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )\mspace{14mu} {and}\mspace{14mu} Y} = ( \frac{V_{c}}{V_{\max}} )}},$

as shown in FIGS. 12 and 13. SE and R² are the standard error andcorrelation coefficient, respectively.Symmetric Tree Model. Equation #16 is also validated in symmetric treesfor various organs and species, as shown in FIG. 14. FIG. 14 shows arelation between

$( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )$

and normalized crown volume in the symmetric vascular tree for variousorgans and species (21-33), corresponding to the table shown in FIG. 10.Parameters B and A are listed in the table shown in FIG. 10, which havea mean±SD value of 1.02±0.02 and 1.00±0.01, respectively, by averagingover various organs and species. These parameters are in agreement withthe theoretical value of one. Furthermore, Equation #15 implies that

${K_{v} = \frac{V_{\max}}{D_{\max}^{2/3}L_{\max}}},$

which can be compared with the regression-derived value. For the tableshown in FIG. 10, parameters B (obtained from least-square fits) and A(obtained from nonlinear regression with B constrained to one) invarious organs when Equation #16 is represented by Equation #17, whereindependent variables

$X = {( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3}}( \frac{L_{c}}{L_{\max}} )}$

and

${Y = ( \frac{V_{c}}{V_{\max}} )},$

as shown in FIG. 14. SE and R² are the standard error and correlationcoefficient, respectively.

FIG. 15 shows a comparison of (K_(v))_(ML) obtained from the nonlinearregression of anatomical data and (K_(v))_(EQ) calculated from Equations#15 and #16. A least-square fit results in a relation of the form:K_(v))_(EQ)=0.998(K_(v))_(ML) (R²=0.999).

Scaling Relations. To further validate the novel volume scaling law ofthe disclosure of the present application, a number of scaling relationsbetween morphological and hemodynamic parameters are provided below. Forthese relations, parameter A has the theoretical value of one asexponent B has a theoretical value of 3/7, 1 2/7, 2⅓ and 3 fordiameter-length relation, volume-length relation, flow-diameterrelation, and volume-diameter relation in Equations #29-32,respectively. The values for A are listed in the table shown in FIG. 11as determined from nonlinear regression. These values, averaged overvarious organs and species, have mean±SD values of 1.01±0.07, 1.00±0.02,0.99±0.05, and 0.99±0.03 for Equations #29-32, respectively. Theagreement of data with theoretical predictions is excellent asdemonstrated by the data referenced herein. For the table shown in FIG.11, the parameter A obtained from nonlinear regression in various organswhen Equations #29-32 (diameter-length, volume-length, flow-diameter,and volume-diameter relations, respectively) are represented by Equation#17. The exponent B is constrained to 3/7, 1 2/7, 2⅓ for 3 for Equations#29-32, respectively. SE and R² are the standard error and correlationcoefficient, respectively.

Volume Scaling Law. Many structural and functional features are found tohave a power-law (scaling) relation to body size, metabolic rates, etc.Previous studies showed several scaling relations connecting structurewith function. A novel volume scaling relation of the disclosure of thepresent application has been demonstrated and validated, which relatesthe crown volume to the stem diameter and crown length.

Clinical techniques (e.g., indicator and dye-dilution method) have beenused to predict blood volume for decades. The blood volume variessignificantly with body size such that it is difficult to evaluate thechange of blood volume in patients because of lack of reference.Although Feldschuh and Enson (Prediction of the normal blood volume:relation of blood volume to body habitus. Circulation. 56: 605-612(1977) used the metropolitan life height and weight tables to determinean ideal weight as an approximate reference, this approach lacks aphysical or physiological basis for calculating normal blood volume. Thenovel volume scaling law of the disclosure of the present applicationmay establish the signature of “normality” and deviation thereof may beindicative of pathology.

The remodeling of intravascular volume may be physiologic during normalgrowth, exercise, or pregnancy. It may also be pathological, however, inhypertension, tumor, or diffuse vascular diseases. Diffuse vasculardisease is difficult to quantify because the normal reference does notexist. The disclosure of the present application shows that the volumescaling law holds in the coronary epicardial trees (vessel diameter>1mm), as shown in FIG. 13 and the table shown in FIG. 9. Such data oncoronaries or other vascular trees are available, for example, byangiography, CT, or MRI. Hence, the novel volume scaling law of thedisclosure of the present application can serve to quantify diffusevascular disease in various organs clinically.

Comparison with ZKM Model. As referenced herein, vascular trees providethe channels to transport fluid to different organs. The optimal designof vascular tree is required to minimize energy losses. Although manytheoretical approaches are proposed to explain the design of vasculartree, the “Minimum Energy Hypothesis” may be the most validatedhypothesis. The ZKM model, based on the minimum energy hypothesis,predicted the exponents

${\chi = \frac{{3ɛ^{\prime}} - 2}{4( {ɛ^{\prime} + 1} )}},{\beta = \frac{5}{ɛ^{\prime} + 1}},{\delta = \frac{4( {ɛ^{\prime} + 1} )}{{3ɛ^{\prime}} - 2}}$

for diameter-length, volume-length, and flow-diameter relations,respectively, where the parameter ε′ in the exponents is the ratio ofmaximum metabolic to viscous power dissipation for a given tree. Basedon Equations #15 and #16 of the disclosure of the present application,the corresponding exponents χ= 3/7, β=1 2/7, and δ=2⅓ are shown. Withthe respective ε′, the mean values over all organs and species are0.43±0.02, 1.28±0.09, and 2.33±0.11 for exponents χ, β, δ, respectively,which agrees well with the present predicted information, i.e.,3/7≈0.43, 1 2/7≈1.29, and 2⅓≈2.33. Furthermore, ZKM model shows themean±SD value of 2.98±0.34 for volume-diameter relation with therespective ε′, which is consistent with the exponent value of 3 inEquation #32. This provides further validation for the proposed volumescaling law of the disclosure of the present application.

Comparison with ¾-power Law. West et al. (A general model for the originof allometric scaling laws in biology. Science. 276:122-126 (1997))proposed the ¾-power scaling law (WBE model) to describe how essentialmaterials are transported in the vascular tree. The WBE model predictsthe following scaling relations: Q_(s)∝M^(3/4), V_(c)∝M, andD_(s)∝M^(3/8). If the first and third relations are combined, oneobtains the flow-diameter relation with an exponent of δ=2, whichimplies that the flow velocity is constant from the large artery to thesmallest arterioles. This is in contradiction with experimentalmeasurements.

If the second and third relations are combined, one obtains thevolume-diameter relation as:

${( \frac{V_{c}}{V_{\max}} ) = {( \frac{D_{s}}{D_{\max}} )^{\frac{8}{3}} = ( \frac{A_{s}}{A_{\max}} )^{\frac{4}{3}}}},$

such that the area-volume relation is

${( \frac{A_{s}}{A_{\max}} ) = ( \frac{V_{c}}{V_{\max}} )^{\frac{3}{4}}},$

where A_(s) and A_(max) are the stem area and the most proximal area,respectively. These WBE predictions differ from the experimentalobservation:

$( \frac{A_{s}}{A_{\max}} ) = {( \frac{V_{c}}{V_{\max}} )^{\frac{2}{3}}.}$

When the cost function in Equation #22 is minimized, one obtains theexponent δ=2⅓, which agrees well with the anatomical data (as shown inthe table of FIG. 10). The area-volume relation

$( {( \frac{A_{s}}{A_{\max}} ) = ( \frac{V_{c}}{V_{\max}} )^{\frac{2}{3}}} )$

obtained from Equation #32 is consistent with the experimentalmeasurements.

There is additional departure of the present model from that of WBE.Equation #30 and V_(c)∝M lead to the following relation:

$\begin{matrix}{L_{c} \propto M^{\frac{7}{9}}} & (18)\end{matrix}$

From Equations #18 and #25, the following relation may be identified:

$\begin{matrix}{Q_{s} \propto M^{\frac{7}{9}}} & (19)\end{matrix}$

From Equation #32 and V_(c)∝M, the following relation may be identified:

$\begin{matrix}{D_{s} \propto M^{\frac{1}{3}}} & (20)\end{matrix}$

Although these scaling relations are different from the WBE model,

$V_{c} \propto {D_{s}^{\frac{2}{3}}L_{c}}$

(Equations #18 and #20 and V_(c)∝M) is still obtained, which furthersupports the validity of Equations #15 and #16. Equation #19 impliesthat the ¾-power scaling law (Q_(s)∝M^(1/4=0.75)) should be 7/9-powerscaling law (Q_(s)∝M^(7/9=0.78)). A least-square fit of Q_(s)−M data hasan exponent value of 0.78 (R²=0.985), which is consistent with the7/9-power scaling law.

Optimal Cost Function. From Equations #26 and #28, the non-dimensionalcost function can be written as follows:

$\begin{matrix}{f_{c} = {{\frac{1}{6}\frac{( {L_{c}/L_{\max}} )^{3}}{( {D_{s}/D_{\max}} )^{4}}} + {( \frac{D_{s}}{D_{\max}} )^{2/3}( \frac{L_{c}}{L_{\max}} )}}} & (21)\end{matrix}$

This is the minimum cost of maintaining an optimal design of a vasculartree under homeostasis. From the structure-function scaling relations(Equation #29),

${\frac{( {L_{c}/L_{\max}} )^{3}}{( {D_{s}/D_{\max}} )^{4}} = {{( \frac{L_{c}}{L_{\max}} )^{1\frac{2}{7}}\mspace{14mu} {and}\mspace{14mu} ( \frac{D_{s}}{D_{\max}} )^{2/3}( \frac{L_{c}}{L_{\max}} )} = ( \frac{L_{c}}{L_{\max}} )^{1\frac{2}{7}}}},$

one may obtain

$\frac{( {L_{c}/L_{\max}} )^{3}}{( {D_{s}/D_{\max}} )^{4}} = \; {( \frac{D_{s}}{D_{\max}} )^{2/3}{( \frac{L_{c}}{L_{\max}} ).}}$

The power required to overcome the viscous drag of blood flow (secondterm in Equation #21) is one sixth of the power required to maintain thevolume of blood (third term in Equation #21). This expression impliesthat most of energy is dissipated for maintaining the metabolic cost ofblood, which is proportional to the metabolic dissipation.

Additional Validation of Volume Scaling Law. From Equations #15 and 16,the disclosure of the present application identifies the cost functionfor a crown, F_(c), consistent with previous formulation.:

F _(c) =Q _(s) ·ΔP _(c) +K _(m) V _(c) =Q _(s) ² ·R _(c) +K _(m) K _(v)D _(s) ^(2/3) L _(c)   (22)

where Q_(s) and ΔP_(c)=Q_(s)·R_(c) are the flow rate through the stemand the pressure drop in the distal crown, respectively, and K_(m) is ametabolic constant of blood in a crown. The resistance of a crown hasbeen identified as

${R_{c} = {K_{c}\frac{L_{c}}{D_{s}^{4}}}},$

where K_(c) is a constant. The cost function of a crown tree in Equation#22 can be written as:

$\begin{matrix}\begin{matrix}{F_{c} = {{Q_{s}^{2} \cdot R_{c}} + {K_{m}K_{v}D_{s}^{2/3}L_{c}}}} \\{= {{K_{c}Q_{s}^{2}\frac{L_{c}}{D_{s}^{4}}} + {K_{m}K_{v}D_{s}^{2/3}L_{c}}}}\end{matrix} & (23)\end{matrix}$

Equation #23 can be normalized by the metabolic power requirements ofthe entire tree of interest, K_(m)V_(max)=K_(m)K_(v)D_(max)^(2/3)L_(max), to obtain:

$\begin{matrix}\begin{matrix}{f_{c} = {\frac{F_{c}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}} =}} \\{= {{\frac{Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}}{( \frac{Q_{s}}{Q_{\max}} )^{2} \cdot \frac{( {L_{c}/L_{\max}} )}{( {D_{s}/D_{\max}} )^{4}}}} + {( \frac{D_{s}}{D_{\max}} )^{2/3}( \frac{L_{c}}{L_{\max}} )}}}\end{matrix} & (24)\end{matrix}$

where ƒ_(c) is the non-dimensional cost function. A previous analysisshows:

$\begin{matrix}{Q_{s} = { {K_{Q}L_{c}}\Rightarrow\frac{Q_{s}}{Q_{\max}}  = \frac{L_{c}}{L_{\max}}}} & (25)\end{matrix}$

where K_(Q) is a flow-crown length constant. When Equation #25 isapplied to Equation #24, the dimensionless cost function can be writtenas:

$\begin{matrix}{f_{c} = {{\frac{Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}} \cdot \frac{( {L_{c}/L_{\max}} )^{3}}{( {D_{s}/D_{\max}} )^{4}}} + {( \frac{D_{s}}{D_{\max}} )^{2/3}( \frac{L_{c}}{L_{\max}} )}}} & (26)\end{matrix}$

Similar to Murray's law, the cost function may be minimized with respectto diameter at a fixed L_(c)/L_(max) to obtain the following:

$\begin{matrix}\begin{matrix}{\frac{\partial f_{c}}{\partial( \frac{D_{s}}{D_{\max}} )} =  0\Rightarrow{\frac{( {- 4} )Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}} \cdot \frac{( {L_{c}/L_{\max}} )^{3}}{( {D_{s}/D_{\max}} )^{5}}} } \\{=  {{- ( \frac{2}{3} )}( \frac{D_{s}}{D_{\max}} )^{\frac{2}{3} - 1}( \frac{L_{c}}{L_{\max}} )}\Rightarrow{\frac{6Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}} \cdot ( \frac{L_{c}}{L_{\max}} )^{2}} } \\{= ( \frac{D_{s}}{D_{\max}} )^{4 + \frac{2}{3}}}\end{matrix} & (27)\end{matrix}$

Equation #27 applies to any stem-crown unit. When L_(c)=L_(max) andD_(s)=D_(max) in Equation #27, one may obtain:

$\begin{matrix}{\frac{6Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}} = { 1\Rightarrow\frac{Q_{\max}^{2}R_{\max}}{K_{m}K_{v}D_{\max}^{2/3}L_{\max}}  = \frac{1}{6}}} & (28)\end{matrix}$

Therefore, Equation #28 can be written as:

$\begin{matrix}{( \frac{D_{s}}{D_{\max}} ) = ( \frac{L_{c}}{L_{\max}} )^{\frac{3}{7}}} & (29)\end{matrix}$

From Equations #16 and #29, one may obtain:

$\begin{matrix}{( \frac{V_{c}}{V_{\max}} ) = ( \frac{L_{c}}{L_{\max}} )^{1\frac{2}{7}}} & (30)\end{matrix}$

From Equations #25 and #29, one may find:

$\begin{matrix}{( \frac{Q_{s}}{Q_{\max}} ) = ( \frac{D_{s}}{D_{\max}} )^{2\frac{1}{3}}} & (31)\end{matrix}$

where Q_(max) is the flow rate through the most proximal stem. FromEquations #29 and #30, one may obtain:

$\begin{matrix}{( \frac{V_{c}}{V_{\max}} ) = ( \frac{D_{s}}{D_{\max}} )^{3}} & (32)\end{matrix}$

Equations #29-32 are the structure-function scaling relations in thevascular tree, based on the “Minimum Energy Hypothesis”. Equations #29,#30, and #32 represent the diameter-length, volume-length, andvolume-diameter relations, respectively and Equation #31 represents thegeneral Murray's law in the entire tree.

The disclosure of the present application also relates to the design andfabrication of micro-fluidic chambers for use in research anddevelopment, thereby designing a chamber that maximizes flow conditionswhile minimizing the amount of material needed to construct the chamber.Many other uses are also possible and within the scope of the disclosureof the present application.

The foregoing disclosure of the exemplary embodiments of the presentapplication has been presented for purposes of illustration anddescription and can be further modified within the scope and spirit ofthis disclosure. It is not intended to be exhaustive or to limit thepresent disclosure to the precise forms disclosed. This application istherefore intended to cover any variations, uses, or adaptations of adevice, system and method of the present application using its generalprinciples. Further, this application is intended to cover suchdepartures from the present disclosure as may come within known orcustomary practice in the art to which this system of the presentapplication pertains. Many variations and modifications of theembodiments described herein will be apparent to one of ordinary skillin the art in light of the above disclosure. The scope of the presentdisclosure is to be defined only by the claims appended hereto, and bytheir equivalents.

Further, in describing representative embodiments of the presentdisclosure, the specification may have presented the method and/orprocess of the present disclosure as a particular sequence of steps.However, to the extent that the method or process does not rely on theparticular order of steps set forth herein, the method or process shouldnot be limited to the particular sequence of steps described. As one ofordinary skill in the art would appreciate, other sequences of steps maybe possible. Therefore, the particular order of the steps set forth inthe specification should not be construed as limitations on the claims.In addition, the claims directed to the method and/or process of thepresent disclosure should not be limited to the performance of theirsteps in the order written, and one skilled in the art can readilyappreciate that the sequences may be varied and still remain within thespirit and scope of the present disclosure.

1.-34. (canceled)
 35. A method for diagnosing disease in a patient'sbiological tree, the method comprising the steps of: obtaining a modelbiological tree; obtaining a patient's biological tree image showing avasculature of at least part of a patient's biological tree; identifyinga length of a patient vessel portion and a diameter of a patient stemvessel proximal to the vessel crown from the patient's biological treeimage; calculating model resistance based upon at least a length of amodel vessel portion and a diameter of a model stem vessel proximal tothe vessel crown from the model biological tree image; calculatingpatient resistance based upon at least the length of the patient vesselportion and the diameter of the patient stem vessel proximal to thevessel crown from the patient's biological tree image; and comparing thecalculated model resistance to the calculated patient resistance todetermine the extent of vessel and/or organ disease.
 36. The method ofclaim 35, wherein the steps of calculating model resistance andcalculating patient resistance are further based upon the use of aconstant.
 37. The method of claim 35, wherein the patient resistance isa resistance of a stem segment, and wherein the length of the patientvessel portion is a length of a stem.
 38. The method of claim 37,wherein the step of calculating the resistance of a stem segment isperformed by dividing the length of a stem by the diameter of thepatient stem vessel proximal to the vessel crown to the fourth powermultiplied by a constant for the stem, or a mathematical equivalentthereof.
 39. The method of claim 38, wherein the constant for the stemis equivalent to one hundred and twenty eight multiplied by a fluidviscosity divided by pi, or a mathematical equivalent thereof.
 40. Themethod of claim 35, wherein the patient resistance is a resistance of acrown, and wherein the length of the patient vessel portion is a crownlength.
 41. The method of claim 40, wherein the step of calculating theresistance of a crown is further based upon the use of a constant forthe crown.
 42. The method of claim 41, wherein the constant for thecrown is equivalent to a maximum resistance multiplied by a maximumdiameter to the fourth power divided by a maximum length, or amathematical equivalent thereof.
 43. The method of claim 40, wherein thestep of calculating the resistance of a crown is performed by dividingthe crown length by the diameter of the patient stem vessel proximal tothe vessel crown to the fourth power multiplied by a constant, or amathematical equivalent thereof.
 44. The method of claim 35, wherein thepatient resistance is a total resistance of the biological tree, andwherein the length of the patient vessel portion is a cumulativebiological tree vessel length.
 45. The method of claim 44, wherein thestep of calculating the total resistance of the biological tree isfurther based upon the use of a constant for the crown.
 46. The methodof claim 44, wherein the step of calculating the total resistance of thebiological tree is performed by dividing the cumulative biological treevessel length by the diameter of the patient stem vessel proximal to thevessel crown to the fourth power multiplied by a constant, or amathematical equivalent thereof.
 47. The method of claim 35, wherein thepatient resistance is the resistance of blood within a vessel portion.48. The method of claim 35, wherein the patient vessel portion comprisesa vessel selected from the group consisting of a blood vessel, a bileduct, a lung, and a neuron.
 49. The method of claim 35, wherein thepatient resistance is the resistance of a gas within a vessel portion.50. The method of claim 49, wherein the gas comprises breathable air,and wherein the vessel portion comprises a portion of a lung.
 51. Themethod of claim 35, wherein the step of comparing the calculated modelresistance to the calculated patient resistance is performed bygraphically comparing patient resistance data to model resistance datato determine the extent of vessel and/or organ disease by identifyingone or more graphical differences between said data.
 52. The method ofclaim 35, wherein the step of comparing calculated model resistance tothe calculated patient resistance is performed by comparing patientresistance data to model resistance data in table form to determine theextent of vessel and/or organ disease by identifying one or morenumerically calculated differences between said data.
 53. The method ofclaim 35, wherein the model biological tree is generated from a minimumenergy hypothesis calculation. 54.-68. (canceled)
 69. A method fordiagnosing disease in a patient's biological tree, the method comprisingthe steps of: obtaining a model biological tree; obtaining a patient'sbiological tree image showing a vasculature of at least part of apatient's biological tree; identifying a diameter of a patient stemsegment and a length of a patient stem segment from the patient'sbiological tree image; calculating model cumulative crown volume basedupon at least a diameter of a model stem segment and a length of a modelstem segment from the model biological tree; calculating patientcumulative crown volume based upon at least the diameter of the patientstem segment and the length of the patient stem segment the patient'sbiological tree image; and comparing the calculated model cumulativecrown volume to the calculated patient cumulative crown volume todetermine the extent of vessel and/or organ disease.
 70. The method ofclaim 69, wherein the step of calculating patient cumulative crownvolume is further based upon the use of a constant.
 71. The method ofclaim 69, wherein the step of calculating patient cumulative crownvolume is performed by multiplying a constant by a product of thediameter of the patient stem segment to the two-thirds power and thelength of the patient stem segment, or a mathematical equivalentthereof.
 72. The method of claim 69, wherein the diameter of the patientstem segment is a most proximal stem diameter, wherein the length of thepatient stem segment is a cumulative biological tree vessel length, andwherein the patient cumulative crown volume is a cumulative biologicaltree volume.
 73. The method of claim 72, wherein the step of calculatingthe cumulative biological tree volume is further based upon the use of aconstant.
 74. The method of claim 72, wherein the step of calculatingcumulative biological tree volume is performed by multiplying a constantby a product of the most proximal stem diameter to the two-thirds powerand the cumulative biological tree vessel length, or a mathematicalequivalent thereof.
 75. The method of claim 69, wherein the step ofcomparing the calculated model cumulative crown volume to the calculatedmodel cumulative crown volume is performed by graphically comparingcalculated model cumulative crown volume data to calculated modelcumulative crown volume data to determine the extent of vessel and/ororgan disease by identifying one or more graphical differences betweensaid data.
 76. (canceled)
 77. The method of claim 69, wherein the stepof comparing the calculated model cumulative crown volume to thecalculated model cumulative crown volume is performed by comparingcalculated model cumulative crown volume data to calculated modelcumulative crown volume data in table form to determine the extent ofvessel and/or organ disease by identifying one or more numericallycalculated differences between said data.
 78. The method of claim 69,wherein the method further comprises the steps of: identifying a mostproximal patient stem diameter from the patient's biological tree image;identifying a patient's cumulative biological tree vessel length fromthe patient's biological tree image; and calculating a patient'scumulative biological tree volume of the patient's biological tree basedupon at least the most proximal patient stem diameter and the patient'scumulative biological tree vessel length.
 79. The method of claim 78,further comprising the step of determining a structure-function scalingrelation, wherein the structure-function scaling relation comprises afirst component and a second component, or a mathematical equivalentthereof.
 80. The method of claim 79, wherein the first componentcomprises the patient's cumulative crown volume divided by the patient'scumulative biological tree volume, and wherein the second componentcomprises the diameter of the patient stem segment divided by the mostproximal patient stem diameter, and wherein the first component equalsthe second component to the third power, or a mathematical equivalentthereof.
 81. The method of claim 79, further comprising the step ofdetermining a non-dimensional structure-function scaling relation,wherein the non-dimensional structure-function scaling relationcomprises a first component, a second component, and a third component,or a mathematical equivalent thereof.
 82. The method of claim 81,wherein the first component comprises the patient's cumulative crownvolume divided by the patient's cumulative biological tree volume,wherein the second component comprises the diameter of the patient stemsegment divided by the most proximal patient stem diameter, wherein thethird component comprises the length of the patient stem segment dividedby the patient's cumulative biological tree vessel length, and whereinthe first component equals the second component to the two-thirds powermultiplied by the third component, or a mathematical equivalent thereof.83. The method of claim 69, wherein the calculated patient's cumulativecrown volume provides information useful for a diagnosis of a disease.84.-121. (canceled)
 122. A system for diagnosing disease in a patient'sbiological tree, the system comprising: a processor; a storage mediumoperably connected to the processor, the storage medium capable ofreceiving and storing data relative of measurements from a vasculatureof a vessel; wherein the processor is operable to: obtain a modelbiological tree; obtain a patient's biological tree image showing avasculature of at least part of the patient's biological tree; identifya length of a patient vessel portion and a diameter of a patient stemvessel proximal to the vessel portion from the patient's biological treeimage; calculate model resistance based upon at least a length of amodel vessel portion and a diameter of a model stem vessel proximal tothe vessel crown from the model biological tree image; calculate patientresistance based upon at least the length of the patient vessel portionand the diameter of the patient stem vessel proximal to the vessel crownfrom the patient's biological tree image; and compare the calculatedmodel resistance to the calculated patient resistance to determine theextent of vessel and/or organ disease.
 123. The system of claim 122,wherein the processor is operable to calculate model resistance andcalculating patient resistance based upon the use of a constant. 124.(canceled)
 125. The system of claim 124, wherein the patient resistanceis a resistance of a stem segment, wherein the length of the patientvessel portion is a length of a stem, and wherein the processor isoperable to calculate the resistance of a stem segment by dividing thelength of a stem by the diameter of the patient stem vessel proximal tothe vessel crown to the fourth power multiplied by a constant for thestem, or a mathematical equivalent thereof.
 126. (canceled) 127.(canceled)
 128. The system of claim 127, wherein the patient resistanceis a resistance of a crown, wherein the length of the patient vesselportion is a crown length, and wherein the processor is operable tocalculate the resistance of a crown based upon the use of a constant forthe crown.
 129. (canceled)
 130. The system of claim 127, wherein theprocessor is operable to calculate the resistance of a crown by dividingthe crown length by the diameter of the patient stem vessel proximal tothe vessel crown to the fourth power multiplied by a constant, or amathematical equivalent thereof.
 131. (canceled)
 132. The system ofclaim 131, wherein the patient resistance is a total resistance of thebiological tree, wherein the length of the patient vessel portion is acumulative biological tree vessel length, and wherein the processor isoperable to calculate the total resistance of the biological tree basedupon the use of a constant for the crown.
 133. The system of claim 131,wherein the processor is operable to calculate the total resistance ofthe biological tree by dividing the cumulative biological tree vessellength by the diameter of the patient stem vessel proximal to the vesselcrown to the fourth power multiplied by a constant, or a mathematicalequivalent thereof. 134.-137. (canceled)
 138. The system of claim 122,wherein the processor is operable to compare the calculated modelresistance to the calculated patient resistance by graphically comparingpatient resistance data to model resistance data to determine the extentof vessel and/or organ disease by identifying one or more graphicaldifferences between said data.
 139. The system of claim 122, wherein theprocessor is operable to compare the calculated model resistance to thecalculated patient resistance by comparing patient resistance data tomodel resistance data in table form to determine the extent of vesseland/or organ disease by identifying one or more numerically calculateddifferences between said data.
 140. (canceled)
 141. The system of claim122, further comprising a program stored upon the storage medium, saidprogram operable by the processor upon data relative of measurementsfrom a vasculature of a vessel,
 142. The system of claim 122, whereinthe system comprises a user system and a server system, and wherein theuser system and the server system are operably connected to one another.143.-159. (canceled)
 160. A system for diagnosing disease in a patient'sbiological tree, the system comprising: a processor; a storage mediumoperably connected to the processor, the storage medium capable ofreceiving and storing data relative of measurements from a vasculatureof a vessel; wherein the processor is operable to: obtain a modelbiological tree; obtain a patient's biological tree image showing avasculature of at least part of a patient's biological tree; identify adiameter of a patient stem segment and a length of a patient stemsegment from the patient's biological tree image; calculate modelcumulative crown volume based upon at least a diameter of a model stemsegment and a length of a model stem segment from the model biologicaltree; calculate a patient's cumulative crown volume based upon at leastthe diameter of the patient stem segment and the length of the patientstem segment the patient's biological tree image; and compare thecalculated model cumulative crown volume to the calculated modelcumulative crown volume to determine the extent of vessel and/or organdisease.
 161. The system of claim 160, wherein the processor is operableto calculate patient cumulative crown volume based upon the use of aconstant.
 162. The system of claim 160, wherein the processor isoperable to calculate patient cumulative crown volume by multiplying aconstant by a product of the diameter of the patient stem segment to thetwo-thirds power and the length of the patient stem segment, or amathematical equivalent thereof.
 163. (canceled)
 164. The system ofclaim 163, wherein the diameter of the patient stem segment is a mostproximal stem diameter, wherein the length of the patient stem segmentis a cumulative biological tree vessel length, and wherein the patientcumulative crown volume is a cumulative biological tree volume, andwherein the processor is operable to calculate the cumulative biologicaltree volume based upon the use of a constant.
 165. The system of claim163, wherein the processor is operable to calculate cumulativebiological tree volume by multiplying a constant by a product of themost proximal stem diameter to the two-thirds power and the cumulativebiological tree vessel length, or a mathematical equivalent thereof.166. The system of claim 160, wherein the processor is operable tocompare the calculated model cumulative crown volume to the calculatedmodel cumulative crown volume is performed by graphically comparingcalculated model cumulative crown volume data to calculated modelcumulative crown volume data to determine the extent of vessel and/ororgan disease by identifying one or more graphical differences betweensaid data.
 167. The system of claim 160, wherein the processor isoperable to compare the calculated model cumulative crown volume to thecalculated model cumulative crown volume by comparing calculated modelcumulative crown volume data to calculated model cumulative crown volumedata in table form to determine the extent of vessel and/or organdisease by identifying one or more numerically calculated differencesbetween said data.
 168. The system of claim 160, wherein the processoris further operable to: identify a most proximal patient stem diameterfrom the patient's biological tree image; identify a patient'scumulative biological tree vessel length from the patient's biologicaltree image; and calculate a patient's cumulative biological tree volumeof the patient's biological tree based upon at least the most proximalpatient stem diameter and the patient's cumulative biological treevessel length.
 169. The system of claim 168, wherein the processor isfurther operable to determine a structure-function scaling relation,wherein the structure-function scaling relation comprises a firstcomponent and a second component, or a mathematical equivalent thereof.170. (canceled)
 171. The system of claim 169, wherein the processor isfurther operable to determine a non-dimensional structure-functionscaling relation, wherein the non-dimensional structure-function scalingrelation comprises a first component, a second component, and a thirdcomponent, or a mathematical equivalent thereof. 172.-173. (canceled)174. The system of claim 160, further comprising a program stored uponthe storage medium, said program operable by the processor upon datarelative of measurements from a vasculature of a vessel.
 175. The systemof claim 160, wherein the system comprises a user system and a serversystem, and wherein the user system and the server system are operablyconnected to one another,
 176. (canceled)
 177. A program having aplurality of program steps to be executed on a computer having aprocessor and a storage medium to analyze data relative of a vasculatureof a vessel, the program operable to: obtain a model biological tree;obtain a patient's biological tree image showing a vasculature of atleast part of the patient's biological tree; identify a length of apatient vessel portion and a diameter of a patient stem vessel proximalto the vessel portion from the patient's biological tree image;calculate model resistance based upon at least a length of a modelvessel portion and a diameter of a model stem vessel proximal to thevessel crown from the model biological tree image; calculate patientresistance based upon at least the length of the patient vessel portionand the diameter of the patient stem vessel proximal to the vessel crownfrom the patient's biological tree image; and compare the calculatedmodel resistance to the calculated patient resistance to determine theextent of vessel disease.
 178. (canceled)
 179. A program having aplurality of program steps to be executed on a computer having aprocessor and a storage medium to analyze data relative of a vasculatureof a vessel, the program operable to: obtain a model biological tree;obtain a patient's biological tree image showing a vasculature of atleast part of a patient's biological tree; identify a diameter of apatient stem segment and a length of a patient stem segment from thepatient's biological tree image; calculate model cumulative crown volumebased upon at least a diameter of a model stem segment and a length of amodel stem segment from the model biological tree; calculate patientcumulative crown volume based upon at least the diameter of the patientstem segment and the length of the patient stem segment the patientsbiological tree image; and compare the calculated model cumulative crownvolume to the calculated model cumulative crown volume to determine theextent of vessel and/or organ disease.